ON GRAPHS WITH THREE DISTINCT LAPLACIAN EIGENVALUES

In this paper,an equivalent condition of a graph G with t(2≤t≤n)distinct Laplacian eigenvalues is established.By applying this condition to t=3,if G is regular(neces- sarily be strongly regular),an equivalent condition of G being Laplacian integral is given.Also for the case of t=3,if G is non-regular,it is found that G has diameter 2 and girth at most 5 if G is not a tree.Graph G is characterized in the case of its being triangle-free,bipartite and pentagon-free.In both cases,G is Laplacian integral.     

Universal inequalities for lower order eigenvalues of self-adjoint operators and the poly-Laplacian

In this paper, we first establish an abstract inequality for lower order eigenvalues of a self-adjoint operator on a Hilbert space which generalizes and extends the recent results of Cheng et al. （Calc. Var. Partial Differential Equations, 38, 409-416 （2010））. Then, making use of it, we obtain some universal inequalities for lower order eigenvalues of the biharmonic operator on manifolds admitting some speciM functions. Moreover, we derive a universal inequality for lower order eigenvalues of the poly-Laplacian with any order on the Euclidean space.     

INEQUALITIES OF EIGENVALUES FOR BI-KOHN LAPLACIAN ON HEISENBERG GROUP

In this article, we consider the eigenvalue problem for the bi-Kohn Laplacian and obtain universal bounds on the （k ＋ 1）-th eigenvalue in terms of the first k eigenvalues independent of the domains.     

Inequalities of Eigenvalues for the Dirac Operator on Compact Complex Spin Submanifolds in Complex Projective Spaces

For a compact complex spin manifold M with a holomorphic isometric embed- ding into the complex projective space,the authors obtain the extrinsic estimates from above and below for eigenvalues of the Dirac operator,which depend on the data of an isometric embedding of M.Further,from the inequalities of eigenvalues,the gaps of the eigenvalues and the ratio of the eigenvalues are obtained.     

EXACT CONTROLLABILITY FOR FIRST ORDER QUASILINEAR HYPERBOLIC SYSTEMS WITH ZERO EIGENVALUES＊＊＊

For a class of mixed initial-boundary value problem for general quasilinear hyperbolic sys-tems with zero eigenvalues, the authors establish the local exact controllability with boundary controls acting on one end or on two ends and internai controls acting on a part of equations in the system.     

The 3D Inverse Problem of the Wave Equation for a General Multi-connected Vibrating Membrane with a Finite Number of Piecewise Smooth Boundary Conditions

The trace of the wave kernel μ（t） =∑ω=1^∞ exp（-itEω^1/2）, where {Eω}ω^∞=1 are the eigenvalues of the negative Laplacian -△↓2 = -∑k^3=1 （δ/δxk）^2 in the （x^1, x^2, x^3）-space, is studied for a variety of bounded domains, where -∞ 〈 t 〈 ∞ and i= √-1. The dependence of μ （t） on the connectivity of bounded domains and the Dirichlet, Neumann and Robin boundary conditions are analyzed. Particular attention is given for a multi-connected vibrating membrane Ω in Ra surrounded by simply connected bounded domains Ω j with smooth bounding surfaces S j （j = 1,……, n）, where a finite number of piecewise smooth Dirichlet, Neumann and Robin boundary conditions on the piecewise smooth components Si^＊ （i = 1 ＋ kj-1,……, kj） of the bounding surfaces S j are considered, such that S j = Ui-1＋kj-1^kj Si^＊, where k0=0. The basic problem is to extract information on the geometry Ω by using the wave equation approach from a complete knowledge of its eigenvalues. Some geometrical quantities of Ω （e.g. the volume, the surface area, the mean curvuture and the Gaussian curvature） are determined from the asymptotic expansion ofexpansion of μ（t） for small │t│.     

DERIVATIVES OF EIGENPAIRS OF SYMMETRIC QUADRATIC EIGENVALUE PROBLEM

Derivatives of eigenvalues and eigenvectors with respect to parameters in symmetric quadratic eigenvalue problem are studied. The first and second order derivatives of eigenpairs are given. The derivatives are calculated in terms of the eigenvalues and eigenvectors of the quadratic eigenvalue problem, and the use of state space representation is avoided, hence the cost of computation is greatly reduced. The efficiency of the presented method is demonstrated by considering a spring-mass-damper system.     

UNIVERSAL BOUNDS FOR EIGENVALUES OF LAPLACIAN OPERATOR OF ANY ORDER

Let Ω be a connected bounded domain in Rn. Denote by λi the i-th eigenvalue of the Laplacian operator with any order p:{0(-△)pu =λu u =■u/→■n =···=■→■np-1 p-u1=0 in Ω,on ■Ω.In this article, we give some expressions for upper bound of the (k + 1)-th eigenvalue λk+1 in terms of the first k eigenvalues.     

由混合谱数据构造矩阵

The eigenvalues and singular values are two of the most distinguished characteristics in a square matrix. Weyl has proved the majorization between them. Horn has proved its inverse, i.e. there exists a matrix with prescribed eigenvalues and singular values. This paper presents a direct transform method which shows the matrix can be upper triangular with its diagonal elements in any order. There exists a real-valued matrix with prescribed complex-conjugate eigenvalues and singular values. Construction of matrices with mixed data is also considered.     

INVERSE PROBLEMS FOR A GENERAL MULTI－CONNECTED BOUNDED DRUM WITH APPLICATIONS IN PHYSICS

this paper studies the influence of a finite container on an ideal gas,The trace of the heat kernel Θ(t)=∑(μ=1)^∞ exp(-tλμ），where{λμ}(μ=1)^∞ are the eigenvalues of the negative Laplacian-△n=-∑(p=1)^n (a/axp)^2 in R^n(n=2 or 3) ,is studied for a general mult-connected bounded drum Ω which is surrounded by simply connected bounded domains Ωi with smooth boundaries aΩi(i=1,……,m) where the Dirichlet ,Neumann and Robin boundary Conditions on aΩi(i=1,……,m) are considered.Some geometrical properties of Ω are determined ,The theremodynamic quantities for an ideal gas encolosed in Ω are examined by using the asymptotic expansions of Θ(t) for short-sime t.It is shown that the ideal gas can not feel the shape of its container Ω，althought it can feel some geometrical properties of it.     

Universal bounds for eigenvalues of the biharmonic operator on Riemannian manifolds

In this paper we consider eigenvalues of the Dirichlet biharmonic operator on compact Riemannian manifolds with boundary (possibly empty) and prove a general inequality for them. By using this inequality, we study eigenvalues of the Dirichlet biharmonic operator on compact domains in a Euclidean space or a minimal submanifold of it and a unit sphere. We obtain universal bounds on the (k+1)th eigenvalue on such objects in terms of the first k eigenvalues independent of the domains. The estimate for the (k+1)th eigenvalue of bounded domains in a Euclidean space improves an important inequality obtained recently by Cheng and Yang.     

Eigenvalues of the buckling problem of arbitrary order and of the polyharmonic operator on Ricci flat manifolds

In this paper, we study eigenvalues of the buckling problem of arbitrary order and of the polyharmonic operator on bounded domains in Ricci flat manifolds supporting a special function and obtain universal bounds on the (k+1)$(k+1)$th eigenvalue in terms of the first k eigenvalues independent of the domains.     

Variation of Grassman powers and spectra

The operator norm of the derivative of the map which takes a finite-dimensional linear operator to its kth Grassman power (the kth compound) is evaluated. This leads to a bound for the distance between the Grassman powers of two operators. As an important application, a bound for the distance between the eigenvalues of two operators is obtained.     

Bound states of a two-boson system on a two-dimensional lattice

We consider a Hamiltonian of a two-boson system on a two-dimensional lattice Z². The Schrödinger operator H(k₁, k₂) of the system for k₁ = k₂ = π, where k = (k₁, k₂) is the total quasimomentum, has an infinite number of eigenvalues. In the case of a special potential, one eigenvalue is simple, another one is double, and the other eigenvalues have multiplicity three. We prove that the double eigenvalue of H(π,π) splits into two nondegenerate eigenvalues of H(π, π ? 2β) for small β > 0 and the eigenvalues of multiplicity three similarly split into three different nondegenerate eigenvalues. We obtain asymptotic formulas with the accuracy of β² and also an explicit form of the eigenfunctions of H(π, π ?2β) for these eigenvalues.     

Error bounds in the isometric Arnoldi process

Error bounds for the eigenvalues computed in the isometric Arnoldi method are derived. The Arnoldi method applied to a unitary matrix U successively computes a sequence of unitary upper Hessenberg matrices H_k, k = 1,2,… The eigenvalues of the H_k's are increasingly better approximations to eigenvalues of U. An upper bound for the distance of the spectrum of H_k from the spectrum of U, and an upper bound for the distance between each individual eigenvalue of H_k and one of U are given. Between two eigenvalues of H_k on the unit circle, there is guaranteed to lie an eigenvalue of U. The results are applied to a problem in signal processing.     

Effective equidistribution of eigenvalues of Hecke operators

In 1997, Serre proved an equidistribution theorem for eigenvalues of Hecke operators on the space S(N,k) of cusp forms of weight k and level N. In this paper, we derive an effective version of Serre's theorem. As a consequence, we estimate, for a given d and prime p coprime to N, the number of eigenvalues of the pth Hecke operator Tp acting on S(N,k) of degree less than or equal to d. This allows us to determine an effectively computable constant Bd such that if J₀(N) is isogenous to a product of Q-simple abelian varieties of dimensions less than or equal to d, then N?Bd. We also study the effective equidistribution of eigenvalues of Frobenius acting on a family of curves over a fixed finite field as well as the eigenvalue distribution of adjacency matrices of families of regular graphs. These results are derived from a general “all-purpose” equidistribution theorem.     

An integral method for solving nonlinear eigenvalue problems

We propose a numerical method for computing all eigenvalues (and the corresponding eigenvectors) of a nonlinear holomorphic eigenvalue problem that lie within a given contour in the complex plane. The method uses complex integrals of the resolvent operator, applied to at least k column vectors, where k is the number of eigenvalues inside the contour. The theorem of Keldysh is employed to show that the original nonlinear eigenvalue problem reduces to a linear eigenvalue problem of dimension k. No initial approximations of eigenvalues and eigenvectors are needed. The method is particularly suitable for moderately large eigenvalue problems where k is much smaller than the matrix dimension. We also give an extension of the method to the case where k is larger than the matrix dimension. The quadrature errors caused by the trapezoid sum are discussed for the case of analytic closed contours. Using well known techniques it is shown that the error decays exponentially with an exponent given by the product of the number of quadrature points and the minimal distance of the eigenvalues to the contour.     

Estimates for eigenvalues on Riemannian manifolds

In this paper, we investigate eigenvalues of the Dirichlet eigenvalue problem of Laplacian on a bounded domain Ω in an n-dimensional complete Riemannian manifold M. When M is an n-dimensional Euclidean space Rn, the conjecture of Pólya is well known: the kth eigenvalue λk of the Dirichlet eigenvalue problem of Laplacian satisfies

     

Bounds for ratios of the membrane eigenvalues

For a bounded planar region in R², we obtain the ratios of lower order eigenvalues of Laplace operator. Combining our results with the recursive formula in Cheng and Yang (2007) [11], we can obtain better upper bound of the (k+1)-th (k?3) membrane eigenvalues.     

An explicit formula for eigenvalues of Bethe trees and upper bounds on the largest eigenvalue of any tree

A Bethe tree B_d,k is a rooted unweighted of k levels in which the root vertex has degree equal to d, the vertices at level j(2?j?k-1) have degree equal to (d+1) and the vertices at level k are the pendant vertices. In this paper, we first derive an explicit formula for the eigenvalues of the adjacency matrix of B_d,k. Moreover, we give the corresponding multiplicities. Next, we derive an explicit formula for the simple nonzero eigenvalues, among them the largest eigenvalue, of the Laplacian matrix of B_d,k. Finally, we obtain upper bounds on the largest eigenvalue of the adjacency matrix and of the Laplacian matrix of any tree T. These upper bounds are given in terms of the largest vertex degree and the radius of T, and they are attained if and only if T is a Bethe tree.